An ideal stiff-string synthesis model is drawn in
Fig. 6.13 [10]. See
§C.6 for a detailed derivation. The delay-line length
is the number of samples in periods at frequency
, where
is the number of the highest partial supported (normally the last
one before
). This is the counterpart of
Fig. 6.12 which depicted ideal-string damping which
was lumped at a single point in the delay-line loop. For the
ideal stiff string, however, (no damping), it is dispersion
filtering that is lumped at a single point of the loop. Dispersion
can be lumped like damping because it, too, is a linear,
time-invariant (LTI) filtering of a propagating wave. Because it is
LTI, dispersion-filtering commutes with other LTI systems in
series, such as delay elements. The allpass filter in
Fig.C.9 corresponds to filter
in Fig.9.2 for
the Extended Karplus-Strong algorithm. In practice, losses are also
included for realistic string behavior (filter
in
Fig.9.2).

Allpass filters were introduced in §2.8, and a fairly
comprehensive summary is given in Book II of this series
[452, Appendix C].7.8The general transfer function for an allpass filter is given (in the
real, single-input, single-output case) by

where
is an integer pure-delay in samples (all delay lines
are allpass filters),

and

We may think of
as the flip of
. For example,
if
, we have
. Thus,
is obtained from
by simply reversing the order of the
coefficients (and conjugating them if they are complex, but normally
they are real in practice). For an allpass filter
simulating
stiffness, we would normally have
, since the filter is already in
series with a delay line.